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  "body": "\n## Summary\n\nThe boundary integral method solves the [[Dirichlet problem|8D56GQ4AG8]] for the Laplace equation with a [[double layer potential|EG8Z3FWV9A]].\nThe unknown harmonic function is written as that potential, with an unknown density on the boundary.\nThe [[jump relations|J58YA8QNNX]] turn the boundary condition into a Fredholm integral equation of the second kind for the density.\nThe [[Fredholm alternative|4HH8C9V8JQ]] then reduces existence of a solution to uniqueness.\nIts discretisation is the boundary element method of numerical analysis.\n\n## Explanations\n\n### L1 human\n\nInstead of looking for a function everywhere inside a region, look only for a function on its boundary.\nSpread unknown dipoles over the boundary; whatever their strengths, they produce a harmonic function inside.\nThen choose the strengths so that this function has the prescribed values on the boundary, which is one equation on the boundary only.\nFor the unit ball in space and boundary value $1$, the constant density $-1$ does the job, and the potential it produces is $1$ throughout the ball.\n",
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